Weighted Compact Nonlinear Scheme Research Articles

High Order Semi-Implicit WCNS for 1D Isentropic and Full Euler Equations

This paper designs a high order semi-implicit weighted compact nonlinear scheme (WCNS) to solve 1D isentropic and full Euler equations. The Euler equations are split into stiff and non-stiff terms that are solved by the implicit and explicit time discretization method, respectively, in order to improve the computational efficiency for low Mach flows. The fifth-order WCNS is applied for the spatial discretization. Several numerical examples are given to demonstrate the performance of the designed method.

Comparison Study of the Fifth-Order WCNS and WENO Scheme In the Truncation Error, Dissipation and Dispersion Terms

This paper makes a comparison study of the fifth-order weighted compact nonlinear scheme (WCNS) and weighted essentially non-oscillatory (WENO) scheme in the truncation error, dissipation and dispersion terms. Truncation error analysis on the WCNS shows that numerical discretization leads to dissipation and dispersion terms. The Fourier spectral analysis methods is used to analyze the dissipative and dispersive features of both schemes.

An Optimised Five-Point-Stencil Weighted Compact Nonlinear Scheme for Hyperbolic Conservation Laws

Although weighted compact nonlinear scheme (WCNS) has been successfully applied to the simulation of complex configurations, due to the excessive dissipation, the scheme is still not suitable for many circumstances. In this article, an optimised five-point-stencil WCNS is proposed. On the one hand, by assembling two fourth-order sub-schemes, the background upwind scheme is optimised for low dissipation. On the other hand, with a specific function, the weighting strategy is optimised to enhance stability near discontinuities and further reduce dissipation in the smooth region. Finally, several benchmark problems are conducted to demonstrate the improved performance of the optimised scheme.

High order semi-implicit weighted compact nonlinear scheme for the all-Mach isentropic Euler system

The computation of compressible flows at all Mach numbers is a very challenging problem. An efficient numerical method for solving this problem needs to have shock-capturing capability in the high Mach number regime, while it can deal with stiffness and accuracy in the low Mach number regime. This paper designs a high order semi-implicit weighted compact nonlinear scheme (WCNS) for the all-Mach isentropic Euler system of compressible gas dynamics. To avoid severe Courant-Friedrichs-Levy (CFL) restrictions for low Mach flows, the nonlinear fluxes in the Euler equations are split into stiff and non-stiff components. A third-order implicit-explicit (IMEX) method is used for the time discretization of the split components and a fifth-order WCNS is used for the spatial discretization of flux derivatives. The high order IMEX method is asymptotic preserving and asymptotically accurate in the zero Mach number limit. One- and two-dimensional numerical examples in both compressible and incompressible regimes are given to demonstrate the advantages of the designed IMEX WCNS.

Numerical simulations on propane/oxygen detonation in a narrow channel using a detailed chemical mechanism: formation and detailed structure of irregular cells

Numerical simulations of two-dimensional inviscid detonations for a stoichiometric propane/oxygen gas mixture are performed using a detailed chemical reaction model. The UC San Diego model which includes 57 chemical species and 268 elementary reactions is mainly used in the present study. It is shown that a grid size of 3 µm can capture important features such as the unburned gas pocket behind the detonation when compared to larger grid sizes. The effects of channel width show that the detonation propagates with the CJ (Chapman–Jouguet) velocity for all cases and for more than 100 times the channel width of 4.5 mm. Increasing the channel width results in an irregular detonation cell structure. A transverse detonation forms with cross-hatching marks on the maximum pressure history. The irregular detonation cell structure forms because both the reduced activation energy and the stability parameter have a value of approximately 10; however, the maximum thermicity in the detonation is one. The free radicals C3H7 and H2O2 play an important role in the propane oxidation under the high temperature in the detonation. The maximum concentration exists at a temperature of 2000–3000 K. The fifth-order WCNS (weighted compact nonlinear scheme) scheme can resolve the contact surface and complicated flow structure behind the detonation front compared to the second-order MUSCL (Monotonic Upstream-centered Scheme for Conservation Laws).

High-Order Hybrid WCNS-CPR Scheme for Shock Capturing of Conservation Laws

By introducing hybrid technique into high-order CPR (correction procedure via reconstruction) scheme, a novel hybrid WCNS-CPR scheme is developed for efficient supersonic simulations. Firstly, a shock detector based on nonlinear weights is used to identify grid cells with high gradients or discontinuities throughout the whole flow field. Then, WCNS (weighted compact nonlinear scheme) is adopted to capture shocks in these areas, while the smooth area is calculated by CPR. A strategy to treat the interfaces of the two schemes is developed, which maintains high-order accuracy. Convergent order of accuracy and shock-capturing ability are tested in several numerical experiments; the results of which show that this hybrid scheme achieves expected high-order accuracy and high resolution, is robust in shock capturing, and has less computational cost compared to the WCNS.

A third‐order compact nonlinear scheme for compressible flow simulations

SummaryReynolds‐averaged Navier‐Stokes simulations based on second‐order numerical methods are widely used by commercial codes and work as dominating tools for most industrial applications. They, however, suffer from limitations in accurate and reliable predictions of skin‐friction drag and aerodynamic heating, as well as in simulations of complex flows such as large‐scale separation and transition. A remedy for this is the development of high‐order schemes, by which numerically induced dissipation and dispersion errors of low‐order schemes can be effectively reduced. Weighted compact nonlinear schemes (WCNSs) are a family of high‐resolution nonlinear shock‐capturing methods. A stencil‐selection procedure is introduced in the proposed work with an aim to improve the nonlinear weight of the third‐order WCNS. By using the approximate dispersion relation analysis, it is demonstrated that the new scheme has reduced dissipation and dispersion errors, compared with WCNSs using two typical nonlinear weights. Improvements are also achieved by the new scheme in numerical tests such as the double Mach reflection problem and the Rayleigh‐Taylor instability simulation, which are characterized by strong shock discontinuities and rich small scales, respectively. The new scheme is therefore highly favored in the simulation of flow problems involving strong discontinuities and multiscales phenomena.

Development of linear unstable modes in supersonic streamwise vortices using a weighted compact nonlinear scheme

Growth processes in supersonic streamwise vortices with linear unstable modes at Mach numbers 2.5 and 5.0 were numerically investigated using a weighted compact nonlinear scheme (WCNS) with three different accuracies. In the evolution of the inviscid linear unstable mode m=−6 as a high-wavenumber mode, the growth rate and eigenfunction profiles of the mode numerically resolved were generally consistent with those obtained with the linear stability theory (LST) during the early transition stage, regardless of the computational accuracy. The numerical scheme used here can capture the growth of three-dimensional linear disturbance in structure. However, the numerical results obtained by nonlinear developments differed according to the accuracy. Among the different accuracies under the same grid resolution, the ninth-order accuracy scheme for the interpolation of primitive variables was able to capture small vortical structures at the downstream in the supersonic flow. In addition, the negative circulation generated and the total disturbance energy indicated that such high accuracy is effective in resolving the developed flow in supersonic vortices, even at moderate grid resolutions.

A multi-resolution weighted compact nonlinear scheme for hyperbolic conservation laws

ABSTRACT A typical weighted compact nonlinear scheme (WCNS) uses a convex combination of several low-order polynomials approximated over selected candidate stencils of the same width, achieving non-oscillatory interpolation near discontinuities and high-order accuracy for smooth solutions. In this paper, we present a new multi-resolution fifth-order WCNS by making use of the information of polynomials on three nested central spatial sub-stencils having first-, third- and fifth-order accuracy, respectively. The new scheme is capable of obtaining high-order spatial interpolation in smooth regions, and it is characterised by the feature of gradually degrading from fifth-order down to first-order accuracy as large stencils deemed to be crossing strong discontinuities. The advantages of the present scheme include the superior resolution for high-wavenumber fluctuations and the flexibility of implementing different numerical flux functions.

A simple extended compact nonlinear scheme with adaptive dissipation control

Weighted compact nonlinear schemes (WCNSs) [1] are a popular family of high resolution shock-capturing methods, of which the nonlinear interpolation procedure is essential for non-oscillatory shock-capturing computation. In this article, numerical dissipation of an improved compact nonlinear scheme, which is fifth-order accurate, is significantly reduced by further introducing a six-point optimized stencil in the nonlinear interpolation, without using more complex local or global smoothness indicators. Nonlinear dissipation and dispersion are then further reduced by using an adaptive strategy. In particular, the computational procedure including the proposed six-point nonlinear interpolation is almost the same as that of the improved five-point scheme, and thus their computational efforts are similar. However, significant gains in numerical performance are obtained while including the six-point optimized stencil. Benchmark cases in one and two space dimensions are used to assess performance of the present scheme, viz., shock-capturing capability and resolution in smooth regions, showing promising advantages.

Development of targeted compact nonlinear scheme with increasingly high order of accuracy

In this paper we present general formulation of targeted compact nonlinear scheme (TCNS) with increasing order of accuracy. This new scheme introduces the incremental-size and ENO-like stencil selection of targeted essentially non-oscillatory (TENO) scheme to weighted compact nonlinear scheme (WCNS). Other aim of the paper is construction of new global smoothness indicators (GSI) for TCNS. We used various benchmark problems of scalar hyperbolic equations and system of Euler equations for investigation of performance of TCNS and new GSIs.

Development of targeted compact nonlinear scheme with increasingly high order of accuracy

In this paper we present general formulation of targeted compact nonlinear scheme (TCNS) with increasing order of accuracy. This new scheme introduces the incremental-size and ENO-like stencil selection of targeted essentially non-oscillatory (TENO) scheme to weighted compact nonlinear scheme (WCNS). Other aim of the paper is construction of new global smoothness indicators (GSI) for TCNS. We used various benchmark problems of scalar hyperbolic equations and system of Euler equations for investigation of performance of TCNS and new GSIs.

High-order maximum-principle-preserving and positivity-preserving weighted compact nonlinear schemes for hyperbolic conservation laws

In this paper, the maximum-principle-preserving (MPP) and positivity-preserving (PP) flux limiting technique will be generalized to a class of high-order weighted compact nonlinear schemes (WCNSs) for scalar conservation laws and the compressible Euler systems in both one and two dimensions. The main idea of the present method is to rewrite the scheme in a conservative form, and then define the local limiting parameters via case-by-case discussion. Smooth test problems are presented to demonstrate that the proposed MPP/PP WCNSs incorporating a third-order Runge-Kutta method can attain the desired order of accuracy. Other test problems with strong shocks and high pressure and density ratios are also conducted to testify the performance of the schemes.

Towards optimal high-order compact schemes for simulating compressible flows

Weighted compact nonlinear schemes (WCNS) (Deng and Zhang, 2000) were developed to improve the performance of the compact high-order nonlinear schemes (CNS) by utilizing the weighting technique originally designed for WENO schemes, and excellent shock capturing capability and high resolution are achieved. Numerous articles have been contributing to improving the performance of WCNSs ever since. In this work, the ENO-like stencil selection procedure of Targeted ENO schemes is introduced for interpolating midpoint variables, targeting compact nonlinear schemes which fully abandon the oscillatory stencils crossing discontinuities, and directly apply optimal linear weights when the flow field is smooth, such that the optimal numerical resolution is fully recovered in smooth flow field. Several canonical numerical cases of scalar equations and the Euler equations of gas dynamics are given to examine the performance of the presented method.

A parameter‐free ε‐adaptive algorithm for improving weighted compact nonlinear schemes

SummaryIn this paper, we propose a parameter‐free algorithm to calculate ε, a parameter of small quantity initially introduced into the nonlinear weights of weighted essentially nonoscillatory (WENO) scheme to avoid denominator becoming zero. The new algorithm, based on local smoothness indicators of fifth‐order weighted compact nonlinear scheme (WCNS), is designed in a manner to adaptively increase ε in smooth areas to reduce numerical dissipation and obtain high‐order accuracy, and decrease ε in discontinuous areas to increase numerical dissipation and suppress spurious numerical oscillations. We discuss the relation between critical points and discontinuities and illustrate that, when large gradient areas caused by high‐order critical points are not well resolved with sufficiently small grid spacing, numerical oscillations arise. The new algorithm treats high‐order critical points as discontinuities to suppress numerical oscillations. Canonical numerical tests are carried out, and computational results indicate that the new adaptive algorithm can help improve resolution of small scale flow structures, suppress numerical oscillations near discontinuities, and lessen susceptibility to flux functions and interpolation variables for fifth‐order WCNS. The new adaptive algorithm can be conveniently generalized to WENO/WCNS with different orders.

An improved consistent, conservative, non-oscillatory and high order finite difference scheme for variable density low Mach number turbulent flow simulation

A consistent, conservative and truly high order finite difference scheme for low Mach number turbulent variable density flow simulation is presented. In the present scheme, the scalar transport equation is discretized with the explicit weighted compact nonlinear scheme (WCNS) while the continuity and momentum equations are discretized with a conservative centered high order finite difference scheme on a staggered grid. Bounded hybrid high order upwind interpolation is proposed for the evaluation of density on the cell faces of the staggered grid. The use of explicit WCNS along with the bounded density interpolation resolves the issue of degraded order of accuracy in the existing high order finite difference schemes where the centered finite difference scheme for mass and momentum conservation is combined with a total variation stable scheme for scalar transport. With the same difference operator used for the continuity and scalar transport equations, the present scheme is consistent naturally, and resolves the inconsistency problem reported in Trisjono et al. (2016) [6] without employing any modification to a scheme. Such improvements are demonstrated for several test cases.

Numerical dissipation control in an adaptive WCNS with a new smoothness indicator

A novel adaptive central-upwind weighted compact nonlinear scheme (WCNS) based on an adjusting parameter of the smoothness indicator is proposed. For the purposes of restraining numerical dissipation in smooth regions, preserving shock-capturing ability and avoiding spurious numerical oscillations around discontinuous regions, an adaptive parameter is introduced in this improved WCNS. The improved scheme can automatically modify the adaptive parameter to fit nonlinear weights to discontinuity according to the local flow-field properties obtained by a discontinuity detector. In smooth regions, the scheme inclines to an optimal central scheme to minimize dissipations and capture turbulent features. While, in discontinue regions, it is more likely to behave as an up-wind scheme for stable shock-capturing ability and numerical robustness. Furthermore, to solve the problem of losing accuracy, a smoothness indicator with a mapping function is introduced. The order-recover property of the improved scheme can obviously avoid unsymmetrical breaking of the flow field. A variety of benchmark cases are tested to verify the improved WCNS performance. The computational efficiencies of all cases have been estimated by measuring the computational time. Numerical results demonstrate that the improved WCNS exhibits excellent shock-capturing ability, lower numerical dissipation, higher numerical robustness and efficiency in resolving complex flow features.

Further improvement of weighted compact nonlinear scheme using compact nonlinear interpolation

To further improve the resolution of weighted compact nonlinear schemes(WCNS), a new 7th-order compact nonlinear interpolation method is proposed on the same stencil as 5th-order CRWENO scheme. Proper nonlinear weights are developed based on the Y type nonlinear weights, the properties of which are further analyzed in this paper. It is found that the Y type nonlinear weights are equivalent to the original nonlinear weights with an adaptive ϵ, which has very small value near discontinuities and very large value in smooth regions. As a result, the new nonlinear scheme has good shock capturing ability and achieves optimal order of accuracy in smooth regions. A new characteristic projection method is put forward, which largely reduces the computation costs. Simulations of Lax problem, Osher-Shu problem, double Mach reflection problem, decay of homogenous turbulence, shock turbulence interaction and turbulent channel flow show that the new scheme captures shocks without obvious oscillations, has higher resolution than other schemes tested, has obvious advantage in distinguishing turbulence from shocks and has higher efficiency.

High-order localized dissipation weighted compact nonlinear scheme for shock- and interface-capturing in compressible flows

Simulations of single- and multi-species compressible flows with shock waves and discontinuities are conducted using a weighted compact nonlinear scheme (WCNS) with a newly developed sixth order localized dissipative interpolation. In smooth regions, the scheme applies the central nonlinear interpolation with minimum dissipation to resolve fluctuating flow features while in regions containing discontinuities and high wavenumber features, the scheme suppresses spurious numerical oscillations by hybridizing the central interpolation with the more dissipative upwind-biased nonlinear interpolation. In capturing material interfaces between species of different densities, a quasi-conservative five equation model that can conserve mass of each species is used to prevent pressure oscillations across the interfaces. Compared to upwind-biased interpolations with classical nonlinear weights [1,2] and improved weights [3], and the interpolation with adaptive central-upwind weights for scale-separation [4], it is shown that WCNS with the proposed localized dissipative interpolation has better performance to simultaneously capture discontinuities and resolve smooth features.

High-order detached-eddy simulation method based on a Reynolds-stress background model

Referring to the construction of shear stress transport-improved delayed detached-eddy simulation (SST-IDDES) method, a variant of IDDES method based on the Speziale-Sarkar-Gatski/Launder-Reece-Rodi (SSG/LRR)-ω Reynolds-stress model (RSM) as Reynolds-averaged Navier-Stokes (RANS) background model, is proposed. Through combining high-order weighted compact nonlinear scheme (WCNS), the SSG/LRR-IDDES method is applied to three aeronautic cases and compared with traditional methods:SST-unsteady Reynolds-averaged Navier-Stokes (URANS), SSG/LRR-URANS, and SST-IDDES. To verify the SSG/LRR-IDDES method in simulating airfoil stalled flow, NACA0012 airfoil is adopted separately at attack angles of 17°, 45° and 60°. At the attack angle of 17°, SST-URANS, SSG/LRR-URANS, and SST-IDDES methods each predict a higher lift coefficient than the experimental data, while the SSG/LRR-IDDES method obtains a better lift coefficient result and a higher fidelity vortical flow structure. It indicates that the RSM can improve the prediction of RANS-mode for pressure-induced separations on airfoil surfaces in detached-eddy simulation. At the attack angles of 45° and 60°, the SSG/LRR-IDDES method captures the massively separated flow with three-dimensional vortical structures and obtains a good result, which is the same as that from the traditional SST-IDDES method. To indicate the improvement of the SSG/LRR-IDDES method in simulating airfoil trailing edge separation, NACA4412 airfoil is adopted. At the attack angle of 12° (maximum lift), the trailing edge separation is mainly induced by pressure gradient. The SSG/LRR-IDDES method can predict the separation process reasonably and obtains a good lift coefficient and location of separation compared with experimental results. However, none of other methods can predict trailing edge separation. It confirms that when RSM is adopted as RANS background model in detached-eddy simulation, the ability to predict pressure-induced separation on airfoil surface is improved. For further verifying the SSG/LRR-IDDES method for simulating three-dimensional separated flow, blunt-edge deltawing at the attack angle of 24.6° is adopted. At this attack angle, the primary vortex will break, which is difficult to predict by using the SST-URANS method. For the SSG/LRR-URANS method, it predicts the vortex breakdown successfully, but the breakdown process does not show any significant unsteady characteristic. The SST-IDDES and the SSG/LRR-IDDES methods both predict a significant unsteady vortex breakdown. But in terms of the accuracy of surface pressure and the fidelity of unsteady flow, the result obtained by the SSG/LRR-IDDES method is better than by the SST-IDDES method.